ICSE Board Sample Paper
Subject – Mathematics
Class – 10th
Time – 2 ½ Hours Full Marks – 80
SECTION – A (40 Marks)
Answer all questions from this section.
1. (a) If x/a = y/b = z/c, prove that each ratio is equal to [(2x3 – 3y3 + 5z3)/(2a3 – 3b3 + 5c3)]1/3. 3
(b) If sin 540 cosec (900 – θ) = 1, find the value of θ, 00 < θ < 900. 3
(c) Construct a rhombus ABCD of side 4.5 cm and LBAD = 60º, by using ruler and compasses only. Draw the lines of symmetry. Hence prove that diagonals are perpendicular to each other. 4
2. (a) Find the amount and compound interest on Rs14000 for 2 tears at 5%. 3
(b) Solve the quadratic equation : x2 – 16 = 0 3
(c) Two chords AB and CD of a circle intersect internally at a point P. If AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD. 4
3. (a) If (a2 + c2)(b2 + d2) = (ab + cd)2, prove that a, b, c, d are in proportion. 3
(b) If x ε R, solve: 2x – 3 ≥ x + (1 – x)/3 > 2/5 x. Also represent the solution on the number line. 3
(c) A shopkeeper buys a mobile phone at a discount of 10% from the wholesaler, the printed price of the mobile phone being Rs4600 and the rate of sales tax is 5%. The shopkeeper sells it to buyer at the printed price and charges tax at the same rate. Find :
(i) the price at which the mobile phone can be bought.
(ii) the VAT paid by the shopkeeper. 4
4. (a) Find the mean, median and mode of the following distribution :
3, 8, 10, 8, 10, 7, 6, 10, 6, 13, 10. 3
(b) List the elements of the solution set of the in equation – 3 < x – 2 ≤ 9 – 2x, x ε N. 3
(c) Without using set square or protractor construct a rhombus ABCD with sides of length 4 cm and one diagonal AC of length 5 cm. Draw its lines of symmetry. Also mark its point of symmetry. 4
SECTION – B (40 Marks)
Answer any four questions from this section.
5. (a) Let A = {1, 2, 3} and a relation on A be R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 1), (3, 3)}. Prove that the relation R is: (i) reflexive, (ii) symmetric, (iii) transitive. 3
(b) Solve the equation : 1/(x + 1) + 2/(x + 2) = 4/(x + 4). 3
(c) Eliminate θ between the equations : x = a cos θ + b sin θ, y = a sin θ – b cos θ. 4
6. (a) Calculate the ratio in which the line joining A(6, 5) and B(4, – 3) is divided by the line y = 2. 3
(b) Two persons standing on the same side of a tower in a straight line with it, measure the angle of elevation of the top of the tower as 30º and 60º respectively. If the height of the tower is 70 m, find the distance between the two persons. 4
(c) A line passes through the point P(3, 2) and cuts off positive intercepts, on the x-axis and the y-axis in the ratio 3 : 4. Find the equation of the line. 3
7. (a) AB is a diameter of a circle with centre O. CD is a chord equal to the radius of the circle. AC and BD produced meet at P. Prove that LAPB = 600. 3
(b) A and B are the points (– 2, 0) and (0, 5). Find the co-ordinates of two points C and D such that ABCD is a square and calculate the length of the diagonal AC. 3
(c) Plot the points A(2, – 3), B (– 1,2) and C(0, – 2) on the graph paper. Draw the triangle formed by reflecting these points in the x-axis. Are the two triangles congruent? 4
8. (a) A man invests Rs3960 in shares of a company which pays 15% dividend at a time when a Rs25 share costs Rs33. Find :
(i) the number of shares he bought
(ii) the annual income from his shares
(iii) the rate of interest which he gets on his investment. 3
(b) If (a3 + 3ab2)/(3a2b + b3) = (x3 + 3xy2)/(3xy2 + y3) , prove that x/a = y/b. 3
(c) Fro the top of a cliff 90 m high, the angle of depression of the top and bottom of a tower are observed to be 300 and 600 respectively. Find the height of the tower. 4
9. (a) Shabana has a cumulative time deposit account in State Bank of India. She deposits Rs500 per month for a period of 4 years. If at the time of maturity she gets Rs28410, find the rate of (simple) interest. 3
(b) If x = a sin θ, y = b tan θ, prove that : a2/x2 – b2/y2 = 1. 3
(c) Mr Sharma has 60 shares of nominal value Rs100 and he decides to sell them when they are at a premium of 60%. He invests the proceeds in shares of nominal value of Rs50 quoted at 4% discount, paying 18% dividend annually. Calculate :
(i) the sale proceeds.
(ii) the number of shares he buys.
(iii) his annual dividend from these shares. 4
10. (a) Draw a circle of radius 3 cm and inscribe a square in it. Measure and record the length of one side of the square drawn. 3
(b) A spherical shell of iron whose internal radius is 9 cm is melted into a conical solid of 28 cm in diameter and 4 3/7 cm height. Find the inner diameter of the shell. [Take π = 22/7] 4
(c) Draw a line segment AB of length 12 cm. Mark M, the mid-point of AB. Draw and describe the locus of a point which is :
(i) at a distance of 3 cm from AB.
(ii) at a distance of 5 cm from the point M. 3